Cause and Correlation in Biology: A User’s Guide to Path Analysis, Structural Equations, and Causal Inference in R. 

Bill Shipley. Cambridge, UK: Cambridge University Press, 2000, 317 pages. 

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Cause and Correlation in Biology: A User’s Guide to Path Analysis, Structural Equations, and Causal Inference in R.

Bill Shipley. Cambridge, UK: Cambridge University Press, 2016

available from Amazon

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This book describes the art of causal modelling.  It is written for biologists (although I expect that others will find it useful as well) and does not assume any statistical background.  As the subtitle suggests, it describes path analysis, structural equations and the fine art of inferring causal relationships from observational data.  You will find a description of what these methods are, how they developed, how to carry out such analyses, and how to interpret the results. 

Here are two other books that discuss similar topics:

Pearl, J. 2000. Causality.  Models, Reasoning, and Inference. Cambridge University Press.

Spirtes, P., Glymour, C. & Scheines, R. 1993. Causation, Prediction, and Search. Springer-Verlag.

Independent reviews of this book

Structural Equation Modeling (2001) 8:646-649

Ecology 82(12), 2001, pp. 3567-3568

Amazon.com

Nature Cell Biology (2002) 4:E37

Biometrics, 2002,  57( No.3), p.989

Annals of Botany, 2002, (december)

Massimo Pigliucci, Univ. of Tennessee, Knoxville, Exec. VP-Society for the Study of Evolution

STRUCTURAL EQUATION MODELING, 8(4), 646, 649

Copyright Ó 2001, Lawrence Erlbaum Associates, Inc.

BOOK REVIEW

Cause and Correlation in Biology: A User’s Guide to Path Analysis, Structural Equations, and Causal Inference. Bill Shipley. Cambridge, UK: Cambridge University Press, 2000, 317 pages, $69.95 (hardcover).

Reviewed by Scott L. Hershberger

There was a time when the number of introductory books on structural equation modeling (SEM) were few. Within the last few years, this has changed considerably; and now it is not difficult to find a reference tailored to one’s background knowledge and expertise. For readers seeking further understanding of SEM, this is a desirable state of affairs, but for authors it has created a new challenge. Authors must now find a way in which to make their writing unique in order to distinguish their work from others. This new book by Bill Shipley takes an approach to SEM that differs considerably from prior introductory volumes.

Virtually all previous volumes have presented SEM using a framework inherited form the Keesling-Wiley-Jöreskog LISREL model, the Bentler-Weeks model, or other similar models. In these models, a model is specified based on theory. This model is then fit to the data. If the model fits, then the model is provisionally accepted; if the model is rejected, a specification search is conducted for the purpose of revising the model so that it eventually fits the data. Frequently, the final model resulting from the specification search differs considerably in structure from the initially hypothesized model. Although the process of fitting a model and its subsequent revision forms a part of the process discussed by Shipley, the process of specifying the initial model differs considerably, owing more to the framework developed by Pearl (2000), glymour, Scheines, Spirtes, and Kelly (1987), and several others. In this later framework, considerable emphasis is placed on how one discovers an optimal model before it is even fit to the data.

According to Shipley, a model implies a series of conditional independence relationships among a set of observed variables. That is, some variables are unrelated when conditioned on some third or more variables. These relationships are reflected in the correlations among the variables. Using the property of d-separation, one can deduce from the correlations which variables, when positioned in a model, should always be conditionally independent. The necessity for the model to mirror this conditional independence will severely restrict the structure of the model. D-separation is by no means the only method one can use to deduce the structure of a model from the correlations among a set of variables. Indeed, late in the book (chapter 8), Shipley introduces the Causal Inference Algorithm, a comprehensive method by which one can specify the exact form of the relation (e.g., X ® Y; X ¬ Y; X « Y) between every pair of variable to be placed in the (acyclic) model. In fact, although never stated as such by Shipley, once one has constructed a model from the relations implied by a set of correlations, excellent model-data fit should be obtained. There should be no need for the laborious specification search that frequently follows the evaluation of an initially hypothesized model, for if one follows the rules and procedures for constructing a model to begin with, relations inconsistent with the correlational pattern will not be incorporated to begin with.

It will not be surprising to many that the beauty and elegance of the model construction procedures discussed by Shipley come at a high intellectual price; that is to say, they’re not easy to understand and learn. Nonetheless, Shipley has done a masterful job of presenting these techniques, and I do not see how it could have been done any better. Chapter 1 deals with the issue of causality, and how causal relations can be inferred from controlled experiments. The author notes that inferring causal relations from controlled experiments is not a completely risk-free endeavor, and that, in some respects, causal relations can be just as securely inferred from correlations. The key to understanding how causality may be inferred from correlational associations rests with the idea of control or statistically holding constant one or more variables. In traditional experiments, the more controls placed on the experiments, the greater the degree of confidence in the causal relationship between the independent and dependent variable. That is, there is still a statistical relationship even though many variables are held constant. This reasoning may be extended to correlational data – one may infer causality by conditioning the relation between two variables on some set of variables. If the relation is still significant, even this evidence consistent with causality. Of course this does not determine in all, or even most, situations the directionality of the relationship (additional subsidiary assumptions and statistical patterns are required), but it is a beginning from which one can move from the "language of causality" (i.e., the path model) to the "language of probability distributions" (i.e. the correlational pattern among the variables). That is, a path diagram implies some relations to be conditionally independent given other variables, and these should be mirrored in the observed correlations. These conditional independence relations are described by the property of d-separation.

D-separation is described in detail in chapter 2. Although, as noted above, d-separation alone provides limited evidence of causality, it is an important beginning to the configuration of a path model. Shipley also sites a number of circumstances under which the accuracy of d-separation is limited: for example, feedback loops, imposed conservation relationships, "unfaithfulness", and context sensitive independence. Chapter 2 is also devoted to defining the many arcane terms to be found in SEM.

Chapter 3 begins with an insightful description of the history of SEM, but is primarily devoted to the significance testing involved in confirming the conditional independence relationship predicted by d-separation. Some of the material in chapter 3 is very traditional; for example, how the significance of a partial correlation may be assessed. Other material is quite new; for example, the d-sep tests developed by Shipley for confirming the structure of a model. Although relatively underdeveloped when compared with more complex model testing procedures such as maximum likelihood, methodologically, d-sep test are very convenient, simple, and based on less restrictive assumptions than other model testing procedures.

In many respects, chapters 4 through 6 contain the material most familiar to structural equation modelers. Chapter 4 outlines the steps that usually comprise SEM: specifying a path diagram representing a model; translating the model into a series of linear, structural equations; deriving the variances and covariances among the observed variables implied by the structural equations; estimating the model parameters using a technique such as maximum likelihood estimation; and evaluating the fit of the model to the data.

Chapter 5 introduces measurements models, and is thus the first place in the book to really deal with latent variables. (The d-sep tests described in chapter 3 are not applicable to models containing latent variables.) Among the strong points of this chapter is a discussion of the consequences of ignoring measurement error, and the limited use of tetrad equations for confirming the existence of latent variables.

The "full" structural equation model is introduced in chapter 6. It must be noted that the material presented in this chapter provides an excellent example of the laudable approach Shipley takes throughout the book. Not content to simplistically present the standard information found in most SEM books, Shipley discusses some of the most difficult yet important issues in SEM: extensive discussions of identification, the consequences of violating the distributional assumptions of maximum likelihood, and measures of model fit based on the noncentral chi-square distribution. And he does so with remarkable clarity! For example, I have never read a clearer rationale of the comparative fit index.

One of the more difficult chapters is chapter 7, which is primarily concerned with using SEM to evaluate hierarchical of multilevel models. However, if one is searching for an introduction to this area, one could do no better than to read this chapter, which presents a convincing rationale for the importance of correctly modeling structural dependencies in one’s data; this dependency can be modeled by using standard SEM software. Undoubtedly, however, the most difficult yet important chapter is chapter 8. Here, the author presents a full exposition as to how, based ont the observed correlations among a set of variables, one can determine the configuration of a model. Although chapter 8 is the last chapter of the book, the procedures described should form the initial steps in SEM. The model construction algorithms contained in this chapter are so complex as to preclude a cursory description of them. Suffice it to say that it is a miracle of human intellect that their development was even possible. It appears that, when properly applied, the vast majority of linear models are resolvable given a correlational pattern among a set of variables. This is not to imply that these model-building algorithms are perfect of fully developed. Much work is still needed in developing algorithms that can lead to nonrecursive models, distinguishing among equivalent models, and models that contain latent variables.

In summary, Cause and Correlation in Biology is the perfect introduction to SEM. The newer SEM framework developed by Pearl and others is coherently combined with the older framework represented by work of Jöreskog, Bentler, and others, to produce a comprehensive introduction to this critical area of statistics. And don’t be put off by the mention of "biology" in the book’s title: Although most of the examples in the book are biologically based, one’s discipline does not have to be so-based to profit intellectually from this text. In fact, the specification of "biology" is irrelevant. This book can be used as the primary text in SEM course given within any discipline, and can be used by scholars and researchers from any area of science.

REFERENCES

Ecology, 82(12), 2001, pp. 3567-3568

AMAZON.COM REVIEW: Path analysis and structural equation modeling, October 17, 2001
Reviewer: ftjfj from Fairbanks, AK United States
This is an excellent guidebook to path analysis and structural equations for biologists. The text is very readable (and even amusing in spots). There is interesting background on the development of path analysis and SEM, as well as thoughtful explanations of how these analyses can link hypotheses of causality to probability theory. New techniques for model testing are introduced that may add additional utility to the statistical method. I found it to be a thoughtful and thorough introduction to a potentially powerful new technique in biology and ecology.

Nature Cell Biology (2002) 4:E37  http://cellbio.nature.com

Between 1920 and 1970, the Dow Jones Industrial Average rose and fell with the lengths of skirts in fashion magazines. As skirts became shorter, stock prices shot higher. Glamorous as this may sound, it does not imply that designers and investors influenced one another – they could also have been subject to the same external cures. In statistical terms, correlation does not imply causation.

Most of us accept this claim as logically indisputable, but we might have to reconsider its practical value. Statistical patterns can allow us to make educated guesses about causal mechanisms that otherwise would remain obscure. The pragmatic question is then not if correlation implies causation, but how. How should experiments be designed and models be tested when we only see the statistical shadows cast by the underlying processes? These questions are sharpened and to some extent answered in Bill Shipley's Cause and Correlation in Biology.

Most experiments in molecular biology are set up so that sophisticated theory is superfluous. By varying one factor and keeping the others as constant as possible, one hopes for straightforward interpretations. Randomization, as when assigning drugs or placebos to subjects of a clinical study, solves the same problem: it can filter out known and unknown factors in one sweep and with a statistical confidence that can be calculated in advance. The real problem arises when your experiments can be neither controlled nor randomized. In some cases the solution is to exercise statistical control by organizing the data set: conditioning on a variable can mimic the effect of physically keeping it constant. For instance, if A and B are independent apart from a shared dependence on C, conditioning on C reveals their independence. By contrast, if A and B both affect C, conditioning on C will make them observationally dependent even if they are causally independent.

The examples above illustrate the idea behind Shipley's D-separation test. Starting with a box-and-arrow causal model, the test derives an exhaustive list of implied conditional independencies. Because independent variables should be uncorrelated, the list can be checked with standard statistical methods. The great appeal of the approach is that virtually all details can be left unspecified. The shapes of distributions are irrelevant and only the existence or absence of causal effects matters. As Shipley proceeds to more sophisticated methods, the scope narrows accordingly. Structural equations modelling (SEM), the main topic of the book, typically assumes linear associations between normally distributed variables. The basic idea is to translate a causal graph into a full probabilistic model, derive the covariances and statistically compare model predictions with observed data. So how is this unusual? Entire disciplines are already occupied with explaining statistical fluctuations in terms of probabilistic mechanisms. Even if few of these models are drawn as graphs, they also reflect causality relations and can be tested statistically. The justification for SEM is that its idealizations give you something in return. Most non-systematic models are developed without descriptions for how they can be rigorously tested, and data is typically handled with little concern for the physical phenomena. SEM takes the unusual and commendable stance of actively merging probabilistic modelling with statistical data analysis. By keeping within modelling limitations, the model is automatically suitable for rigorous statistical tests, including systematic treatments of unmeasured variables, measurement errors and hierarchically structured data. The book concludes with algorithms for discovering models that fit experiments. The strategy is now the inverse. Known statistical correlations are used to generate a list of conditional independencies that, using the notion of d-separation, in turn can be used to infer causal graphs. An inconclusive outcome of the algorithm indicates that an unmeasured variable has a significant influence on a process, something that can also be tested statistically. The methods can, therefore, not only track down models in a given set of variables, but also detect hidden variables.

The methods presented are relevant in all fields of biology, but perhaps not as urgently as in ecology and biometrics where Shipley picks his examples. Most cell- or molecular-level experiments are designed to look at averages or qualitative features, whereas statistics is used mainly to estimate measurement errors. This could be about to change. Probabilistic modelling is up-and-coming and high-throughput methods are generating a wealth of statistical data. There will then be a greater demand for evaluation tools, but the future use of SEM is still not obvious. Much of theoretical biology focuses on dynamics, and probabilistic models are often formulated using birth-and-death processes or stochastic differential equations. The book barely mentions time and instead presents a static picture where variables are expressed as sums of other variables. Shipley points out that static models could describe equilibrium states, but even these are often more attainable from dynamic models where the change is set to zero.

Addressing students and practising biologists, Shipley does a terrific job of making mathematical ideas accessible. He introduces all methods from scratch, walks through realistic examples and generously supplies historical and philosophical notes. The writing style is overall enthusiastic and playful, but some parts feel half-baked with overly detailed anecdotes. What the book lacks in austerity, it makes up for by an unusual intellectual curiosity and diversity. Cause and Correlation in Biology is a nontechnical and honest introduction to statistical methods for testing causal hypotheses. ÿ

Biometrics (2002), Vol 57 ( No.3), p.989.



Although causal analyses of non-experimental data are primarily employed
by social scientists, the person who was one of earliest proponents and
innovators was the biologist Sewall Wright. Experimentalists, whether they
are biologists or psychologists like myself, are inherently skeptical of
causal inference from non-experimental data. Shipley brings this healthy
skepticism to a discussion of path analysis, structural equation modeling,
and causal inference. He presents an intelligent introduction to these
topics very often including biological illustrations. Those who are
unfamiliar with these techniques and seek to learn about them would benefit
from reading this book.
Path analysis was invented by the aforementioned Sewall Wright, and his
contributions are extensively reviewed in the book. This method typically
involves causal models in which measured standardized variables have causal
relationships, called path coefficients, which are estimated by solving for
them from the observed correlations. Most of the models in path analysis,
though not all, use measured not latent variables, and the solutions for
the parameters are algebraic, not statistical. Although Wright's
contributions were largely ignored by biologists, they were
enthusiastically disseminated by social scientists in the 1960s and 70s,
most notably by the sociologist O. D. Duncan.
Within contemporary social sciences, there is very little data analysis
using path analysis; rather most of the modeling is Structural Equation
Modeling (SEM). In essence, this type of modeling is path analysis with
latent and measured variables. SEM gives causal modeling a more solid
statistical foundation. Experimentalists, as well as statisticians, tend
to be dubious about the concept of latent variables. Likely, some of this
skepticism was due to atheoretical use of exploratory factor analysis.
Those with a background of psychometrics are more accepting of the concept
of a latent variable. The fundamental question in a latent variable
analysis is whether the measured variable exactly corresponds to the
theoretical construct. Biologists obviously realize that the phenotype is
not the genotype, and allowances for measurement error are often necessary.
The book spends a good deal of time discussion of the topic of causal
inference, a topic that had grown out of fashion but has undergone a
recent revival. Unlike the other topics of the book, there are few examples.
One of the attractive features of the book is the literate and readable
text. All too often, a book like this is filled with equations. While
appreciative, I would like to have seen more definitions of technical
terms. For instance, identification and several other concepts are not
defined with enough detail. Additionally, one chapter, "Nested and
Multilevel Models" has insufficient detail. Moreover, the discussion of
multiple groups in that chapter seems out of place.
Reviews of books, this one included, all too often nit-pick and
over-criticize the book because the reviewer did not write the book. The
methods discussed in this book will be increasingly used in biology. To be
a complete biologist, one will need to know these techniques and this book
is the "Wright" place to start.

David A. Kenny

James B. Grace

Cause and correlation in biology.

As indicated by the title, this book seeks to address the relationship between correlation and causation, with application to biological topics. The book begins with a statement of three objectives: (1) to persuade biologists that it is possible to infer causation with observational (non-experimental) data; (2) to describe certain methods that can assist in this process; and (3) to illustrate the methods presented using biological examples.

A philosophical discussion of causation is potentially confusing and can often evoke debate, which necessitates some carefully crafted effort at the beginning of the book to address the meaning of causation. Fortunately for all involved, the methods presented do not depend on any one particular definition of causality, but instead, on certain properties of causal systems, most of which are not subject to much disagreement. To some extent, the practical result of the introductory material is to get the reader accustomed to the word ‘causation’, which requires some effort because the science of statistics typically trains us to avoid its use.

In the first chapter of the book (‘Preliminaries’), the author tackles the task of convincing the reader that there is a substantial need for methods of applying causal inference in the absence of randomized or controlled experiments. To a great extent this is a fairly easy task since virtually everyone, scientist and non-scientist alike, develops causal interpretations of system behaviour as part of the routine business of living. Nonetheless, because this book is about both the philosophy and practice of inferring causal relationships, there is value in pointing out the limits that exist to applying experimental approaches to a great variety of important questions.

The second chapter tackles a topic that, at least formally, is uncharted territory for most biologists: the interrelationship between causality and statistical relationships. The language of directed graphs (a.k.a. path models) is dealt with first. Because of the recent emergence of graphical modelling as a distinct statistical methodology, it is unfortunate that the author did not provide a link to the rather extensive literature on this topic (cf Whittaker, 1990). Most of the chapter deals with the concept of ‘d-separation’ (short for directed separation), which formally defines the logical relationships that can exist among variables in a directed graph. This topic is necessarily dry and even tedious because it is an exhaustive exercise in the logic of all possible relations. However, the author does an excellent job of trying to make this presentation as interesting and informative as possible—not an easy task.

In the third chapter, the author first gives an interesting discussion of the early history of path models and the contrasting viewpoint of experimental statistics that has since prevailed in the biological sciences. The bulk of the chapter, however, is devoted to a presentation of the author’s own statistical methodology for testing the validity of path models. Here the author makes a sincere effort to present in understandable terms an alternative method to the usual maximum likelihood approach for testing path models.

Chapters 4 to 7 present the fundamentals of the methodology of structural equation modelling. Starting with path models and maximum likelihood estimation, the author proceeds through a discussion of measurement error and latent variables, model fit assessment, and multilevel models. These topics are covered for completeness and in order to demonstrate their applicability to biological examples. Currently, there are literally dozens of general treatments of structural equations. Yet, since the method is virtually unknown among biologists, there is justification in including this material in the book. In these chapters the author shows both the essence of structural equation modelling and the very broad range of questions it can address. In the final chapter the author tackles the somewhat specialized field of search strategies for exploring hypothesis space. Except for the author’s own work, this method has not been applied before to biological systems and deserves further scrutiny.

To a significant extent, this book seeks to swing the historical pendulum of biological thought away from a suspicion of linking correlation and causation towards an embrace of causal interpretation of correlational data. This is not an easy task, and it can be said that the author has done a fairly good job of hitting the level of sophistication needed to make the case forcefully and yet coherently. The author’s dedication to the topic of path model analysis is evidenced by his own development of ‘d-sep’ test methods as an alternative to traditional maximum likelihood methods. Time will tell whether this new approach will prove to be adopted by practitioners.

Overall, I found this book to be filled with useful presentations. It provides a good device for exposing biologists to many statistical methods that have gained widespread acceptance in other fields but that have escaped the attentions of biometricians. Like the author, I share an enthusiasm for the potential applications of these methods for exploring the natural world.

It is perhaps useful to the potential reader to point out that there are alternative ways of looking at the value of the methods discussed in this book. The author has chosen to emphasize the search for causality as a motivating force behind the application of path model methods. I, on the other hand, have used many of the same methods but for a different reason: the belief that multivariate analyses yield more insights about complex systems than do bivariate analyses. Actually, multivariate hypotheses formulated as path models can be evaluated with experimental data (e.g. Gough and Grace, 1999) as well as with non-experimental data. Furthermore, the philosophical requirements for establishing causality are precisely the same for a simple bivariate regression as for a complex path model, namely directional dependence. What is different in the two cases (aside from the complexity of showing that data match the model) is the degree of insight gained about the workings of the system. Thus, it is possible to value path modelling methods strictly because they provide hypotheses that more closely match the complexity of natural systems rather than based on any belief that the statistical methods contribute any insights into the question of causal relationships.

Alternative perspectives aside, I highly recommend the book by Shipley for those interested in multivariate approaches to biology. The material is ultimately quite important, the presentation is unusually clear, and the exposure to methods developed in other fields of science is valuable. I would also urge those with a serious interest in the subject to keep an eye out for the forthcoming book by Pugesek et al. (due February 2003) for an extended description of the concepts and applications of structural equation modelling to biological systems. Together, these books are likely to finally break down the barriers that have isolated biologists from path modelling methods for so long.

LITERATURE CITED

Gough, Grace. 1999. Title. Ecology 80: 882–890.

Pugesek et al. Title. Cambridge: Cambridge University Press (in press).

Whittaker J. 1990. Graphical models in applied multivariate statistics. New York: Wiley.

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